Question

c）The price of a European call that expires in six months and has a strike price of $30 is $2. The underlying stock price is $29, and a dividend of $0.50 is expected in two months and in five months. The term structure is flat, with all risk-free interest rates being 10% per year continuously compounded. What is the price of a European put option that expires in six months and has a strike price of $30?

d）Explain carefully the arbitrage opportunities in Problem c) if the European put price trades at $3.

just answer question d

Answer 1

Stirke Price (K) = $30

Stock Price (S) = $29

Risk Free Rate (r) = 10% p.a. Continuously Compounded

Time to Maturity = 6 Months

Dividend 1 (D1) = $0.5 in 2 Months

Dividend 2 (D2) = $0.5 in 5 Months

Value of Call Option = $2

Value of Put Option = $3

We can construct 2 portfolio which would have same cashflow at the maturity. This is similar to arguement in Put - Call Parity Theory. It can be done as follows:

1. Long Stock and Long Put Option

2. Long Call Option and Invest Present Value of Strike Price at risk free rate.

Since payoff is same, the cost of making that portfolio should also be the same. If there is difference, there is an arbitrage opportunity available.

Cost of Making 1st Portfolio:

Long Stock Purchase Price = $29

Purchase Price of Put Option = $3

Present Value of Dividends to be received:

Months Remaining Dividend PV Factor Present Value of Dividends

2 $0.5 0.983 $0.492

5 $0.5 0.959 $0.480

Present Value of Dividends $0.97

Net Cost of Making 1st Portfolio = $29 + $3 - $0.97 =$31.03

Cost of Making 2ndPortfolio:

Purchase Price of Call Option = $2

Strike Price = $30

Present Value of Stirke Price = Strike Price *

Present Value of Stirke Price = 30 *

Present Value of Stirke Price = $28.64

Net Cost of Making 2nd Portfolio = $28.64 + $2 =$30.64

Since we can construct one portfolio cheaply than other one, we can buy portfolio which is cheap and sell portfolio which is expensive to gain out of this mispricing. Investor needs to take following steps:

1. For buying 2nd portfolio:

a. Invest $28.64 in risk free rate for 6 months and receive $30 after 6 months.

b. Buy the Call Option on stock with the strike price of $30 at a rate of $2.

So, totally we need $30.64 now to buy this portfolio.

1. For selling 1st portfolio:

a. Sell the stock for $29 today.

b. Sell the Put Option on stock with the strike price of $30 at a rate of $3.

c. Invest the present value of dividends which need to be paid since we have shorted the stock. For that we need to invest $0.49 in 2 months risk free bonds and $0.48 in 5 months risk free bonds. This way proceeds from these investment would be used in repayment of dividend.

Totally we would have $31.03 by selling this portfolio out of which $30.64 would be used to fund purchase of 2nd portfolio. Remaining would be our arbitrage profit.